Answer
The values are
$x=-2$
$y=-1$
and $z=-2$
The solution set is $\{(-2,-1,-2)\}$.
Work Step by Step
The given equations are
$2x-y+z=-5$ ... (1)
$x-2y-3z=6$ ... (2)
$x+y-2z=1$ ... (3)
Multiply equation (1) by $3$.
$6x-3y+3z=-15$ ... (4)
Add equation (2) and (4).
$x-2y-3z+6x-3y+3z=6-15$
$7x-5y=-9$ ... (5).
Now multiply equation (2) by $2$.
$2x-4y-6z=12$ ... (6)
And multiply equation (3) by $-3$.
$-3x-3y+6z=-3$ ... (7)
Add equations (6) and (7).
$2x-4y-6z-3x-3y+6z=12-3$
$-x-7y=9$
Multiply by $7$.
$-7x-49y=63$ ... (8)
Add equations (5) and (8).
$7x-5y-7x-49y=-9+63$
Simplify.
$-54y=54$
$y=-1$.
Plug the value of $y$ into equation (5).
$7x-5(-1)=-9$
$7x+5=-9$
$7x=-9-5$
$7x=-14$
$x=-2$
Plug the values of $x$ and $y$ into equation (1).
$2(-2)-(-1)+z=-5$
$-4+1+z=-5$
$-3+z=-5$
$z=-5+3$
$z=-2$.
